Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free boundary problem of low Mach number magnetohydrodynamic flows

Ruixi Zhang

Abstract

In this paper we consider a free boundary problem of low Mach number magnetohydrodynamic flow in spatial dimension n $\geq$ 2. A priori estimates of the second fundamental form and various flow quantities in Sobolev norms are obtained by adopting the geometrical point of view introduced by Christodoulou and Lindblad [3]. Moreover, a blow up criterion is derived by using the method of Beale, Kato and Majda [2].

Free boundary problem of low Mach number magnetohydrodynamic flows

Abstract

In this paper we consider a free boundary problem of low Mach number magnetohydrodynamic flow in spatial dimension n 2. A priori estimates of the second fundamental form and various flow quantities in Sobolev norms are obtained by adopting the geometrical point of view introduced by Christodoulou and Lindblad [3]. Moreover, a blow up criterion is derived by using the method of Beale, Kato and Majda [2].

Paper Structure

This paper contains 23 sections, 49 theorems, 370 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Geometric aspects
  5. Commutator estimates
  6. Proof of Theorem \ref{['thm:main']}
  7. Estimate of the temporal derivative of $E_r^c$
  8. Estimates of temporal derivatives of $E_r^a,E_r^b$
  9. Completion of the proof
  10. Regularity estimates I
  11. Boundary regularity
  12. Estimates of $(v,B,q,\mathcal{T})$
  13. Estimates of $\mathop{\mathrm{div}}\nolimits v$ and further regularity
  14. Proof of Theorem \ref{['thm:buc']}
  15. An integral inequality for $E_3^{c\pm}$
  16. ...and 8 more sections

Key Result

Theorem 1.3

Suppose that $(\mathcal{D}_t,v,B,q,\mathcal{T})$ is a smooth solution to (eq:LM) and that the initial data satisfies Then there is a $T=T(\epsilon_{0*},\epsilon_{1*},K_*,L_*,\mathop{\mathrm{vol}}\nolimits\mathcal{D}_0,\mathcal{E}_{r}(0))>0$ such that the following estimates hold for $t\in[0,T]$: and

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 42 more